Centre No. Candidate No. Paper Reference(s) 6664/01 Edexcel GCE Core Mathematics C2 Advanced Subsidiary Friday 24 May 2013 Morning Time: 1 hour 30 minutes Materials required for examination Mathematical Formulae (Pink) Paper Reference 6 6 6 4 0 1 Surname Signature Items included with question papers Nil Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation or symbolic differentiation/integration, or have retrievable mathematical formulae stored in them. Instructions to Candidates In the boxes above, write your centre number, candidate number, your surname, initials and signature. Check that you have the correct question paper. Answer ALL the questions. You must write your answer for each question in the space following the question. When a calculator is used, the answer should be given to an appropriate degree of accuracy. Initial(s) Examiner s use only Team Leader s use only Question Number 1 2 3 4 5 6 7 8 9 10 Blank Information for Candidates A booklet Mathematical Formulae and Statistical Tables is provided. Full marks may be obtained for answers to ALL questions. The marks for individual questions and the parts of questions are shown in round brackets: e.g. (2). There are 10 questions in this question paper. The total mark for this paper is 75. There are 32 pages in this question paper. Any pages are indicated. Advice to Candidates You must ensure that your answers to parts of questions are clearly labelled. You should show sufficient working to make your methods clear to the Examiner. Answers without working may not gain full credit. This publication may be reproduced only in accordance with Pearson Education Ltd copyright policy. 2013 Pearson Education Ltd. Printer s Log. No. P41859A W850/R6664/57570 5/5/5/6/6/ *P41859A0132* Total Turn over
1. The first three terms of a geometric series are respectively, where p is a constant. 18, 12 and p Find (a) the value of the common ratio of the series, (b) the value of p, (1) (1) (c) the sum of the first 15 terms of the series, giving your answer to 3 decimal places. (2) 2 *P41859A0232*
2. (a) Use the binomial theorem to find all the terms of the expansion of (2 + 3x) 4 Give each term in its simplest form. (4) (b) Write down the expansion of (2 3x) 4 in ascending powers of x, giving each term in its simplest form. (1) 4 *P41859A0432*
3. f(x) = 2x 3 5x 2 + ax + 18 where a is a constant. Given that (x 3) is a factor of f(x), (a) show that a = 9 (b) factorise f(x) completely. (2) (4) Given that g(y) = 2(3 3y ) 5(3 2y ) 9(3 y ) + 18 (c) find the values of y that satisfy g(y) = 0, giving your answers to 2 decimal places where appropriate. (3) 6 *P41859A0632*
Question 3 continued *P41859A0732* 7 Turn over
4. y = 5 2 ( x + 1) (a) Complete the table below, giving the missing value of y to 3 decimal places. x 0 0.5 1 1.5 2 2.5 3 y 5 4 2.5 1 0.690 0.5 (1) y 5 R O 3 x Figure 1 Figure 1 shows the region R which is bounded by the curve with equation y the x-axis and the lines x = 0 and x = 3 = 5 2 ( x + 1), (b) Use the trapezium rule, with all the values of y from your table, to find an approximate value for the area of R. (4) (c) Use your answer to part (b) to find an approximate value for 3 0 4 + 5 + 1 dx 2 ( x ) giving your answer to 2 decimal places. (2) 10 *P41859A01032*
Question 4 continued Q4 (Total 7 marks) *P41859A01132* 11 Turn over
5. D E 12 m 0.64 rad A 23m B 12 m C Figure 2 Figure 2 shows a plan view of a garden. The plan of the garden ABCDEA consists of a triangle ABE joined to a sector BCDE of a circle with radius 12m and centre B. The points A, B and C lie on a straight line with AB = 23m and BC = 12m. Given that the size of angle ABE is exactly 0.64 radians, find (a) the area of the garden, giving your answer in m 2, to 1 decimal place, (4) (b) the perimeter of the garden, giving your answer in metres, to 1 decimal place. (5) 12 *P41859A01232*
Question 5 continued *P41859A01332* 13 Turn over
6. y C A O B x Figure 3 Figure 3 shows a sketch of part of the curve C with equation y = x(x + 4)(x 2) The curve C crosses the x-axis at the origin O and at the points A and B. (a) Write down the x-coordinates of the points A and B. (1) The finite region, shown shaded in Figure 3, is bounded by the curve C and the x-axis. (b) Use integration to find the total area of the finite region shown shaded in Figure 3. (7) 16 *P41859A01632*
Question 6 continued *P41859A01732* 17 Turn over
7. (i) Find the exact value of x for which log 2 (2x) = log 2 (5x + 4) 3 (4) (ii) Given that log a y + 3log a 2 = 5 express y in terms of a. Give your answer in its simplest form. (3) 20 *P41859A02032*
8. (i) Solve, for 180 x < 180, tan(x 40 ) = 1.5 giving your answers to 1 decimal place. (3) (ii) (a) Show that the equation can be written in the form sin tan = 3cos + 2 4cos 2 + 2cos 1 = 0 (3) (b) Hence solve, for 0 < 360, sin tan = 3cos + 2 showing each stage of your working. (5) 22 *P41859A02232*
Question 8 continued *P41859A02332* 23 Turn over
9. The curve with equation has a stationary point P. y = x 2 32 x) x Use calculus (a) to find the coordinates of P, (6) (b) to determine the nature of the stationary point P. (3) 26 *P41859A02632*
10. C y 9 O x Figure 4 The circle C has radius 5 and touches the y-axis at the point (0, 9), as shown in Figure 4. (a) Write down an equation for the circle C, that is shown in Figure 4. (3) A line through the point P(8, 7) is a tangent to the circle C at the point T. (b) Find the length of PT. (3) 30 *P41859A03032*
Question 10 continued Q10 (Total 6 marks) END TOTAL FOR PAPER: 75 MARKS 32 *P41859A03232*